The hot spots conjecture for simply connected planar domains

The explanation

Take a thin metal plate with insulated edges, so no heat escapes. Heat it unevenly and wait. The temperature evens out, but slowly, and for a long time it keeps one basic shape. The question is where the hottest and coldest points of that lasting shape sit. The hot spots conjecture says they sit on the edge, never inside.

A rectangle 2 units long and 1 unit wide shows the idea. Its slowest-fading temperature shape is cos(πx/2), which runs from hot at one short end to cold at the other. Its decay rate is π²/4, about 2.47. The next shape across the width decays at π², about 9.87, so the lengthwise one lasts longest. Its hottest and coldest points lie on the two short ends. A rectangle has corners, so it is only a warm-up for the smooth shapes in this paper.

A round plate is smooth. There the lasting shape is a Bessel function times cos θ. For a disc of radius 1, its decay rate is about 3.39. It grows steadily from the centre outward, so again the extremes lie on the rim. A disc also has a twist: the same shape turned by 90 degrees fades at exactly the same rate. Then there is a whole family of lasting shapes, not just one, and each must be checked.

In mathematical terms, these lasting shapes are eigenfunctions for the first positive Neumann eigenvalue. Jeffrey Rauch raised the problem in 1975. It is false in general. In 1999, Krzysztof Burdzy and Wendelin Werner built a flat counterexample with two holes. Burdzy later needed only one hole. So the conjecture is about shapes with no holes. Proofs existed only for special classes. These include convex shapes with two mirror symmetries, the so-called lip domains, and every triangle.

This paper proves a strict form for every smooth flat shape without holes. Every lasting shape has a nonzero slope at every inside pointSource 1. So it has no peak, no valley and no saddle point inside. Its maximum and minimum lie on the edge. This holds for every member of the family when there is more than oneSource 2.

The single difficulty is that the slope has no preferred directionSource 3. In a rectangle you can watch the slope along its length. A general shape gives you nothing like that. The insulated edge does give one fact: at the boundary, the slope runs along the edge, never across itSource 3. The paper builds an energy that is zero exactly for slopes of lasting shapesSource 4. It then reweights the boundary values in a way that keeps that energy zero. If some inside point had zero slope, these reweightings would create too many independent lasting shapes. That breaks a limit the paper proves on how many there can be, so no such point existsSource 5.

The result needs a smooth edge, so it does not cover shapes with corners, such as polygons. It also stays in two dimensions and assumes no holes. The counterexamples show the claim fails for some shapes with holes.

How sure is this? A computer proof checker called Lean has checked a formal version of the main theorem. The slope is never zero inside, for every lasting shape in the family. A tool called Comparator matches it against a fixed reference statement. No specialist review of the paper has been published.

What it doesn't show. This does not cover shapes with corners, shapes with holes, or higher dimensions.